THE PRODUCT THEOREM FOR PARAMETRIZED TOPOLOGICAL REIDEMEISTER TORSION WOJTEK DORABIALA AND MARK W. JOHNSON Abstract.The goal of this article is to prove the product formula for pa* *rametrized topological Reidemeister torsion. The theorem states that the product of* * the parametrized Euler characteristic of one fibration with the parametrized* * Reide- meister torsion class of another fibration yields the parametrized Reide* *meister torsion class of the product fibration. In the process of establishing t* *he the- orem, several new products must be defined involving (derivative theorie* *s of) parametrized A-theory and a detailed description of the coassembly map f* *or parametrized A-theory is included. 1.Introduction Before moving into the statements of the main results, we would like to say a* * few words about our motivation for studying this question. The definition of topolo* *gical Reidemeister torsion used here is based on that of [5]. They conjecture that th* *eir definition should become equivalent to that of [7] once their class is pushed i* *nto cohomology using the Borel regulator technique. In fact, we would eventually li* *ke to establish that our definition, the definition from [7] and the definition from * *[1] agree in cohomology, which would generalize the work of [12] in the non-parametrized case. In [9], the authors give a completely algebraic analog of our product theorem. They establish the fact that the Reidemeister torsion of the tensor product A * * B where B is contractible is equal to the Euler characteristic of A times the tor* *sion of B. Our goal is to give a topological lifting of this result to Whitehead spa* *ces. Moving into specifics, all fibrations considered here will be perfect fibrati* *ons. That is, the fibers of every fibration mentioned will be finitely dominated by * *as- sumption. We will also assume the base space B comes equipped with an effi- cient triangulation, where each simplex contains only finitely many subcomplexe* *s. Clearly, this allows any differentiable compact manifold as a choice of B. The first major result is an A-theory product formula for Euler characteristi* *cs, relying upon the external parametrized A-theory product `E1 ' ` E2 ' ` E1xE2 ' ~A : A # p1^ A # p2! A # p1x p2. B1 B2 B1xB2 Theorem 1.1. Suppose p1 and p2 are perfect fibrations. Then ~A(ØA (p1), ØA (p2)) = ØA (p1 x p2). ____________ Date: March 3, 2003. 1991 Mathematics Subject Classification. Primary: 19D10; Secondary: 18F25, 1* *9Exx, 55R70. Key words and phrases. Reidemeister torsion, parametrized A-theory, parametr* *ized Euler characteristic, homotopy limit, Whitehead space, perfect fibration, retractive * *space. 1 2 W. DORABIALA AND M. W. JOHNSON In order to simplify notation, if a fibration p : E ! B is equipped with a ch* *oice of bundle of finitely generated, free R-modules OE : V ! E, it will be referred* * to as a fibration with flat bundle . The phrase fibration with acyclic flat bundle* * will then imply the additional condition that H*(Eb; Vb) is zero for each b 2 B. For* * the purposes of our desired product formula for parametrized Reidemeister torsion, * *the following corollary is actually the key result. Corollary 1.2. Suppose p1 : E1 ! B1 is a perfect fibration with flat bundle and p2 : E2 ! B2 is a perfect fibration with acyclic flat bundle. Then ~Acy(ØA (p1), ØAcy(p2)) = ØAcy(p1 x p2). The`next'result relies upon the existence of a restricted external multiplica* *tion E for # p: B ` E1 ' ` E2 ' ` E1xE2 ' _cy: # p1^ cy # p2 ! cy # p1x p2. B1 B2 B1xB2 Theorem 1.3. Suppose p1 : E1 ! B1 is a perfect fibration with flat bundle and p2 : E2 ! B2 is a perfect fibration with acyclic flat bundle. Then _cy(Ø (p1), Øcy(p2)) ' Øcy(p1 x p2) where ' means there exists a natural path connecting these points. In section 6, we will establish the existence of a restricted product pairing ` E1 ' ` E2 ' ` E1xE2 ' : # p1^ Wh R2B2# p2 ! Wh R1BAR21xB2 # p1x p2 B1 B2 B1xB2 ` E2 ' where Wh R2B2# p2 represents the Whitehead space associated to p2. Recall that B2 the topological Reidemeister torsion øR2 (p2)may be viewed as a point in this Whitehead space. The main result here is the product formula for torsion which generalizes a classical result of Kwun and Szczarba in [9]. Theorem 1.4 (Product Formula). Suppose p1 : E1 ! B1 is a perfect fibration with flat bundle and p2 : E2 ! B2 is a perfect fibration with acyclic flat bundle. T* *hen øR1 AR2 (p1 x p2)is defined and (Ø (p1), øR2 (p2)) ' øR1 AR2 (p1 x p2). Remark 1.5. Note that in the informal notation of [5], øR1 AR2 (p1 x p2)and (Ø (p1), øR2 (p2)) both correspond to maps B1x B2 ! Wh R1BAR21xB2(E1 x(E2)this is a different version of Whitehead space). The statement of the product formula then becomes that these two maps are homotopy equivalent. The Product Theorem is deduced as a consequence of Theorem 1.3 in section 2. The basic idea, again using the informal notation of [5], is that the follo* *wing diagram commutes up to homotopy. AcyB;(E); ØAcy(p)xxx|x xxx | xxx fflffl| B øR(p)//_WhRB(E) PRODUCT THEOREM FOR PARAMETRIZED REIDEMEISTER TORSION 3 The authors would like to express their thanks to Bruce Williams for helpful discussions concerning this material. 2. The Product Formula This section is devoted to reducing the proofs of Theorems 1.1, 1.3, 1.4 and Corollary 1.2 to the existence of certain products, which will be defined in se* *ction 6, and several technical results which will be proven later. The products we need will be the following. ` E1 ' ` E2 ' ` E1xE2 ' A # p1 ^ A # p2 __~A__//_A # p1x p2 B1 B2 B1xB2 ` E1 ' ` E2 ' ` E1xE2 ' A # p1^ Acy # p2 ~Acy//_Acy # p1x p2 B1 B2 B1xB2 ` E1 ' ` E2 ' ` E1xE2 ' # p1^ # p2 ______//_ # p1x p2 B1 B2 B1xB2 `E1 ' ` E2 ' _cy ` E1xE2 ' # p1 ^ cy # p2_____// cy # p1x p2 B1 B2 B1xB2 ` E1 ' ` E2 ' ` E1xE2 ' # p1 ^ Wh R2B2# p2 _____//WhR1BAR21xB2# p1x p2 B1 B2 B1xB2 Also in Sections 5 and 6, we will construct natural öc assembly maps" ` E ' ` E ' A # p __ffi_//_ # p B B ` E ' ` ' fficy cy E Acy # p _____// # p B B and ` E ' ffl ` E ' cy # p _____//WhRB # p . B B We will define Ø (p)as the image under the coassembly map ffi of the parametr* *ized A-theory Euler characteristic ØA (p)and similarly for Øcy(p). At the end of Sec* *tion 6, this will lead to a proof of the following. Proposition 2.1. Suppose p2 : E2 ! B2 is a perfect fibration with acyclic flat bundle. Then øR2 (p2)= ffl2(Øcy(p2)) In Sections 6 and 7 and we will establish the following key technical result. 4 W. DORABIALA AND M. W. JOHNSON Proposition 2.2. The following diagrams are each commutative up to a natural homotopy: ` E1 ' ` E2 ' ` ' cy cyE1xE2 (1) A # p1^ Acy # p2 ~A__//_A # p1x p2 B1 B2 B1xB2 ffi1xfficy2|| fficy1,2|| ` ' fflffl|`' ` fflffl|' E1 E2 E1xE2 # p1^ cy # p2 __cy_// cy # p1x p2 B1 B2 _ B1xB2 and ` E1 ' ` E2 ' _cy ` E1xE2 ' (2) # p1 ^ cy # p2 _________//_ cy # p1x p2 B1 B2 B1xB2 1xffl2|| |ffl1,2| ` ' fflffl|` ' `fflffl| ' E1 E2 R R E1xE2 # p1 ^ Wh R2B2# p2 _____//Wh1BA12xB2 # p1x p2. B1 B2 B1xB2 In fact, the first statement of the propostion follows immediately from the f* *ol- lowing theorem together with Lemma 2.5 below. Notice, we intend the product rather than smash product versions of ~A and _ for technical reasons. Theorem 2.3. The diagram ` E1 ' ` E2 ' ` E1xE2 ' A # p1 x A # p2__~A_//A # p1x p2 B1 B2 B1xB2 ffi1xffi2|| ffi1,2|| ` ' fflffl|`' ` fflffl|' E1 E2 E1xE2 # p1 x # p2 _____// # p1x p2. B1 B2 _ B1xB2 commutes up to a natural homotopy. We can now establish Theorem 1.1 and Corollary 1.2, which generalize classical properties of the Euler characteristic. Recall the fold map E t E ø E `E ' defines an object of Retfd(p), hence a point in A # p which we denote by ØA (* *p) B and think of as the sphere bundle of the trivial line bundle over E. One might think of this as a parametrized version of the Euler characteristic of the fibe* *r Eb Since ~A is defined as the map induced by the bi-exact functor (by restriction from [16]) external smash product of retractive spaces ^E1xE2 : Retfd(p1)x Retfd(p2)! Retfd(p1 x p2), and the Euler characteristic is represented by E t E ø E in Retfd(p), Theorem 1.1 is a consequence of the following simple lemma, which * *we prove in the next section. PRODUCT THEOREM FOR PARAMETRIZED REIDEMEISTER TORSION 5 Lemma 2.4. One has the identity (E1 t E1)^E1xE2 (E2 t E2)= (E1 x E2)t (E1 x E2) in Retfd(p1 x p2). In order to prove Corollary 1.2, we need the following result on the relation* *ship between ~A and ~Acy. Lemma 2.5. (1) The diagram ` E1 ' ` E2 ' ` E1xE2 ' A # p1^ Acy # p2 ~Acy//_Acy # p1x p2 B1 B2 B1xB2 | | | | ` ' fflffl|`' ` fflffl|' E1 E2 E1xE2 A # p1 ^ A # p2 __~___//_A # p1x p2 B1 B2 A B1xB2 commutes. ` E ' `E ' (2) The canonical map Acy # p ! A # p is an injection for every fibra- B B tion p. ` ' ` ' E E (3) The canonical map cy # p ! # p is an injection for every fibra- B B tion p. Proof of Corollary 1.2.The corollary follows from Theorem 1.1 and Lemma 2.5 along`with'the`fact'that ØAcy(p2)goes to ØA (p2)under the canonical inclusion E2 E2 Acy # p2 ! A # p2. B2 B2 This suffices to allow us to prove Theorems 1.3 and 1.4. Proof of Theorem 1.3.The assertion is that _cy(Ø (p1), Øcy(p2)) ' Øcy(p1 x p2). which by our definition of Ø (p1)and Øcy(p2)is equivalent to _cy(ffi1(ØA (p1)), fficy2(ØAcy(p2))) ' fficy1,2(ØAcy(p1 x p2)). By diagram 1 in Proposition 2.2, it suffices to establish ~Acy(ØA (p1), ØAcy(p2)) = ØAcy(p1 x p2) which is the statement of Corollary 1.2. Proof of Theorem 1.4.The assertion is that (Ø (p1), øR2 (p2)) ' øR1 AR2 (p1 x p2). However, Proposition 2.1 implies this is equivalent to the statement (Ø (p1), ffl2(Øcy(p2))) ' ffl1,2(Øcy(p1 x p2)). Now Proposition 2.2 (2) implies this statement follows immediately from Theorem 1.3. 6 W. DORABIALA AND M. W. JOHNSON A proof of Lemma 2.4 and some basic results on retractive spaces make up Sect* *ion 3. Section 4 is devoted to defining all of the relevant theories, parametrized * *Euler characteristics and torsion. The technical details of Thomason homotopy limit problem maps necessary for constructing the coassembly maps and understanding their basic properties are handled in Section 5. The bulk of Section 6 is devot* *ed to defining the remaining products and establishing their properties, with proo* *fs of Proposition 2.1 and the second statement of Proposition 2.2 at the end. Finally* *, in Section 7 we give the proofs of Lemma 2.5 and of Theorem 2.3. 3.Functors on Retractive Spaces In this section we would like to introduce the external smash product of retr* *ac- tive spaces over a fibration along with two "change of base" functors related to restriction to a subcomplex of the base space B. The external smash products induce the external product in parametrized A-theory and their interaction with the change of base functions is important to the proof of Proposition 2.2. We begin by defining the category of retractive spaces over p, Retfd(p), as t* *he Waldhausen category of retractive spaces X over E where the composition X ! E ! B is a (Hurewicz) fibration whose fibers are finitely dominated. (See [17].* *) The Waldhausen category structure has weak equivalences the homotopy equivalences which happen to be maps in this category and similarly cofibrations are the map* *s in this category which, as continuous maps, are closed embeddings with the homotopy extension property. Recall that a retractive space over E is a pair X ø E where the inclusion i : E ! X is a cofibration and ri = 1E . In this notation, Retfd(* *E) consists of finitely dominated retractive spaces over E, since it assumes the b* *ase B = *. In particular, notice Retfd(p)is a sub-Waldhausen category of Retfd(E). Next, we introduce the external smash product. Definition 3.1. Let X 2 Retfd(p1)and Y 2 Retfd(p2). Then their external smash product is a space in Retfd(p1 x p2)which we define as the following pushout X x E2 [ E1 x Y_______//_X x Y | | | | fflffl| fflffl| E1 x E2_________//X ^E1xE2 Y. Proof of Lemma 2.4.Both terms can be identified with the pushout of the followi* *ng diagram. E1 x (E2 t E2) [ (E1 t E1) x E2__//_E1 x E2 | | | | fflffl| fflffl| (E1 t E1) x (E2 t E2)__________//P Let Simp(B) denote the category of simplices of the chosen triangulation of t* *he base space B, where the morphisms are only the inclusions of subsimplices. If ffi,oe 2 Simp (B), let iffioe: |ffi| ! |oe| and ioeB: |oe| ! B denote the natur* *al inclusions. Notice that iffioeis a homotopy equivalence, hence ~iffioe: p-1(ffi) ! p-1(oe) * *will also be a homotopy equivalence. PRODUCT THEOREM FOR PARAMETRIZED REIDEMEISTER TORSION 7 fd fd Definition 3.2. Given W 2 Retfdp-1(ffi), Y 2 Ret p-1(oe) and X 2 Ret (E) we define pullbacks and pushforwards as below. (1) The pushforward of W along iffioeis the following pushout ~iffioe p-1(ffi)____//p-1(oe) | | | | fflffl| fflffl| W ______//_iffioe*(W ) where ~iffioe: p-1(ffi) ! p-1(oe) is the induced inclusion (which is a h* *omotopy equivalence). (2) Similarly, we can define ioeB*(Y ) as the pushout of Y along the induced inclusion ~ioeB. (3) The pullback of Y along iffioeis the following pullback * iffioe(Y_)____//Y | | | | fflffl|~iffifflffl|oe p-1(ffi)____//p-1(oe) where ~iffioe: p-1(ffi) ! p-1(oe) is the induced inclusion (which is a h* *omotopy equivalence). (4) Similarly, we can define ioeB*(X) as the pullback of X along the induced inclusion ~ioeB. To see that the first two operations land in the correct categories, see [17]. For the pullback operations, it is important to note the fact that iffioeis a* * closed em- bedding implies the fibers of iffioe*(X) are simply certain fibers of X itself* * (thereby preserving the finitely dominated condition). A result of Lück [10] implies the* * in- clusion being a cofibration is preserved by the pullback operation as well. Fin* *ally, the fibration condition is preserved under pullbacks. The following lemma is largely a consequence of the universal properties of pushouts and pullbacks. Lemma 3.3. Given ffi oe, the pullback and pushforward over iffioeyield an adj* *oint pair of functors i * j iffioe*: Retfdp-1(ffi)! Retfdp-1(oe), iffioe: Retfdp-1(oe) ! Retfdp-1(ffi) * Hence, there is a natural homotopy equivalence iffioe*iffioe(Y ) ! Y called th* *e unit of adjunction and similarly for Z ! iffioe*iffioe*(Z). The functor iffioe*is * *an exact functor, while the maps of spaces iffioe*(Y ) ! Y and Z ! iffioe*(Z) are homo* *topy equivalences. Finally, the functors iffioe*and ioeB*preserve homotopy equival* *ences. Notice, the fact that X ! oe or X ! B is assumed to be a fibration implies the pullback preserves homotopy equivalences. We now have a result about the interaction between change of base and external smash products. fd Lemma 3.4. Suppose Wn 2 Retfdp-1n(ffin)and Xn 2 Ret p-1n(oen)with ffin oen for n = 1,2. Then there are natural homotopy equivalences 8 W. DORABIALA AND M. W. JOHNSON (1) * * iffi1oe1(X1) ^p-11(ffi1)xp-12(ffi2)iffi2oe2(X2) | | fflffl| (iffi1oe1x iffi2oe2)*(X1 ^p-11(oe1)xp-12(oe2)X2) (2) (iffi1oe1x iffi2oe2)*(W1 ^p-11(ffi1)xp-12(ffi2)W2) | | fflffl| iffi1oe1*(W1) ^p-11(oe1)xp-12(oe2)iffi2oe2*(W2) Proof.We will display the second map in detail, while the first is dual. To beg* *in, note there is a commutative diagram W1 xOW2O______________________//iffi1oe1*(W1)OxO iffi2oe2*(W2) | | | | | | W1 x p-12(ffi2) [ p-11(ffi1)_x/W2/_iffi1oe1*(W1) x p-12(oe2) [ p-11(oe1) x iff* *i2oe2*(W2) | | | | fflffl| fflffl| p-11(ffi1) x p-12(ffi2)______________//_p-11(oe1) x p-12(oe2) where the horizontal maps are all homotopy equivalences and both maps to the top row are cofibrations. Thus, there is an induced homotopy equivalence on pushouts ffi ffi W1 ^p-11(ffi1)xp-12(ffi2)W2 ! ioe11*(W1) ^p-11(oe1)xp-12(oe2)ioe22*(W2) which factors through an induced map ffi * * ffi (iffi1oe1x iffi2oe2)*(W1 ^p-11(ffi1)xp-12(ffi2)W2) ! ioe11*(W1) ^p-11(oe1)xp-* *12(oe2)ioe22*(W2) by construction. Since W1 ^p-11(ffi1)xp-12(ffi2)W2 ! (iffi1oe1x iffi2oe2)*(W1 ^p-11(ffi1)xp-12* *(ffi2)W2) is also a homotopy equivalence, the statement follows. 4.Parametrized Theories, Euler Characteristics and Torsion Classes ` E ' Definition 4.1. Let A # p = |wS. Retfd(p)| where Retfd(p)is the Wald- B hausen category of retractive spaces over E with the fibers of the composition * *pr finitely dominated, where r : X ! E is the retraction. Definition 4.2. Suppose p is a perfect fibration with flat bundle V . Then ` E ' Acy # p = |wS. Retfd,cy(p)| B PRODUCT THEOREM FOR PARAMETRIZED REIDEMEISTER TORSION 9 where Retfd,cy(p)is the full sub-Waldhausen category of Retfd(p)consisting of t* *hose objects X ø E where r*(V ) ! X makes each fiber Xb ! * a perfect fibration with acyclic flat bundle. Definition 4.3. Suppose p is a perfect fibration with flat bundle V . Let ` E ' AR # p = |wS. RetfdR(p)| B where RetfdR(p)is the category Retfd(p)with a different Waldhausen category str* *uc- ture where the weak equivalences are the local homology chain equivalences with local coefficients in r*V and cofibrations are as usual. (See [4] for details.) `E ' Recall Acy # p is homotopy equivalent to the homotopy fiber of the map B induced by the identity, (which is an exact functor in this direction) `E ' ` E ' A # p ! AR # p B B as in [4]. ` E ' Definition 4.4. We set # p = holim |wS. Retfdp-1(oe)|. B oe2Simp(B) Let f~ W _____//V | | | | fflffl|fflffl| X __f__//Y be a pullback diagram where X is equipped with the flat bundle W and Y is equipped with the flat bundle V . Then we will say (f; ~f) is a bundle morphism from (X; W ) to (Y ; V ). Let D denote the category whose objects are pairs (X;* * W ) where X has flat bundle W and morphism set D ((X; W )(Y ; Vt))he set of bundle maps (f, ~f) as above. We would like to think of Acy as a functor from D to T op. Thus, we need to understand the map Acy(X; W )! Acy(Y ; V )induced by a bundle morphism (f, ~f). We define a functor f* from Retfd(X)to Retfd(Y )by taking Z ø X to Z[X Y ø Y . We would like to know that, restricting the source to the full subcategory Retfd,cy(X)(W), the target lies in the full subcategory Retfd,cy(Y()V.)This req* *uires a statement from homological algebra; specifically, H*(Z, X; W ) = 0 must imply H*(f*(Z), Y ; V ) = 0. However, this follows from the definition of homology wi* *th local coefficients. (See [3].) Now we would like to know the functor f* is exact. See Lemma 3.3 for the fact that f* is exact in our case when considered as a functor f* : Retfd(X)! Retfd(Y ). However, since both cyclic categories are sub-Waldhausen categories, the exactn* *ess in our case follows by restriction. Thus, f* induces a map Acy(X) ! Acy(Y ). Now we define a functor P : Simp(B) ! D associated to any fibration p with fl* *at bundle V (so (E; V ) 2 D) by sending oe 7! (p-1(oe); V |p-1(oe)). Any morphism * *ffi ! oe 10 W. DORABIALA AND M. W. JOHNSON in Simp (B)is the inclusion of a subcomplex. Thus, the horizontal rectangles and right squares in the following diagram are pullbacks, which implies the left sq* *uares are pullbacks as well. V |p-1(ffi)_//V |p-1(oe)_//V | | || | | | |fflffl fflffl| fflffl| p-1(ffi)____//p-1(oe)___//_E | | | | | | |fflffl fflffl| fflffl| ffi_________//_oe_____//B Hence, we have defined a morphism (p-1(ffi); V |p-1(ffi)) ! (p-1(oe); V |p-1(oe)) associated to each morphism ffi ! oe, thereby making our assignment P a functor. We can now define the composite functor cy Simp (B) _P__//_D_A__//T op and this is the functor whose holimappears in the following definition. ` E ' Definition 4.5. We set cy # p = holim |wS. Retfd,cyp-1(oe)|. B oe2Simp(B) Replacing the functor Acy : D ! T op with the functor AR : D ! T op we may form a similar holim. ` E ' Definition 4.6. We set R # p = holim |wS. RetfdRp-1(oe)|. B oe2Simp(B) `E ' Definition 4.7. Let ØA (p)2 A # p denote the point corresponding to B E t E ø E as an object of Retfd(p), with the fold map as retraction. Definition`4.8.'Suppose p is a fibration with acyclic flat bundle V . Let ØAcy(* *p)2 E Acy # p denote the point corresponding to EtE ø E as an object of Retfd,cy(p). B Notice that H*((E t E)b, Eb; Vb) is naturally isomorphic to H*(Eb, Vb) = 0, so the assumption of acyclic flat bundle is simply a restatement of the fact that * *the sphere bundle of the trivial`line'bundle, E t E ø E, is an object of Retfd,cy(p* *). E Now define Ø (p)2 # p as the image of ØA (p)under the coassembly map B ffi we will describe in detail in section 5. (This agrees with the definition o* *f Ø (p) given in [5].) Similarly, set Øcy(p)= fficy(ØAcy(p)). Next, we define the parametrized Whitehead space associated to a fibration p with flat bundle V . Notice the inclusions p-1(oe) ! E will yield a map ` E ' # p = holim A p-1(oe)! holim A (E). B Simp(B) Simp(B) PRODUCT THEOREM FOR PARAMETRIZED REIDEMEISTER TORSION 11 Follow this by the map holimA (E) ! holim K (R) Simp(B) Simp(B) induced by the linearization map associated to V . Denote the composition ` E ' V : # p ! holim K (R). B Simp(B) `E ' Definition 4.9. Let Wh RB # p denote the homotopy fiber of V. B For the sake of precision, we should mention our model for K(R) is |wS.Ch*(R* *) | where Ch*(R) is the category of bounded complexes of projective modules given a Waldhausen structure where injections are cofibrations and quasi-isomorphisms (homology equivalences) are the weak equivalences. Then the linearization map is induced by the functor Retfd(E) ! Ch*(R) sending X to the relative chain complex with local coefficients and compact supports C*(X, E; r*(V )). It will be shown at the end of the next section that there is a canonical pat* *h fi in holim K (R) from V(Ø (p)) to the basepoint provided the flat bundle over p * *is Simp(B) assumed to be acyclic. Definition 4.10. Suppose p is a`fibration'with acyclic flat bundle V . Then the E Euler characteristic Ø (p)2 # p together with the path fi define a point in ` ' B E Wh RB # p by definition of homotopy fiber. We will refer to this point as the B torsion class øR (p)or the topological Reidemeister torsion. 5. Thomason Homotopy Limit Problems We would like to understand the definition of the coassembly map `E ' ` E ' ffi : A # p ! # p. B B This comes from a formal trick which travels under the name of a Thomason ho- motopy limit problem which we will describe in an abstract context. In order to* * be careful with the properties of the realization of a category, in this section o* *nly, we will make explicit the composition of the nerve functor and the geometric reali* *zation of a simplicial set. Suppose C is a small category and G : C ! Cat is a functor. There is always another important functor C ! Cat, defined by C 7! C=C sending each object in C to the category of objects over it in C. Clearly this is a covariant functor, d* *efined on morphisms by postcomposition. Given two functors with the same source and target, one can define the category of natural transformations between them, wh* *ich we'll denote by Nat (C=?, G). Recall that one normally views a natural transformation as a point in a large product of mapping sets satisfying certain additional conditions. More precise* *ly, natural transformations lie in the equalizer of the two maps induced by precomp* *o- sition and postcomposition respectively. (See section IX.5 of [11] for the tran* *slation as an "end".) 12 W. DORABIALA AND M. W. JOHNSON Thus, it is natural to define Y Nat (C=?, G) F un(C=C, G(C)) C2C as the equalizer in Cat of two functors. The first functor Y Y F un(C=C, G(C)) ! F un(C=C, G(D)) C2C ':C!D2C is induced by the collection of functors G(')* : F un(C=C, G(C)) ! F un(C=C, G(D))'. The second functor is defined similarly using the precomposition, (C=')* : F un(C=D, G(D)) ! F un(C=C, G(D))'. In other words, a natural transformation is a collection of functors (C) : C=C ! G(C) where the following diagram commutes for each morphism ' : C ! D in C: (C) C=C _____//_G(C) C='|| |G(')| fflffl| fflffl| C=D _(D)//_G(D). Similarly, a morphism in the category Nat (C=?, G)consists of a natural transfo* *r- mation in each factor which must be compatible in the sense that a certain cubi* *cal extension of this diagram commutes. The nerve functor is a right adjoint, hence it preserves equalizers. Thus, ap* *plying nerve to Nat (C=?, G)yields an equalizer in the category of simplicial sets. Ta* *king the geometric realization does not preserve arbitrary products, but does preser* *ve equalizers. Hence, there is a natural isomorphism |N(Nat (C=?, G))| eq(|N( )|, |N( )|). However, one must keep in mind that |N( )| and |N( )| no longer decompose as the products of |N(?)| applied to the expected factors. We let ~ and ~ denote the same constructions where the product is taken after the realization and the components are of the form |N(F un(C=C, G(C)))| and |N(F un(C=C, G(D)))|. Notice there is a natural map |N( )| ! ~ given by the universal property of the product in the definition of ~ and similarly for . Now suppose there is another small category D together with a functor F : D ! Nat (C=?, G). Then taking nerves yields a natural map N(F) : N(D) ! N(Nat (C=?, G)) and taking geometric realization gives a natural map |N(D)| ! |N(Nat (C=?, G))| PRODUCT THEOREM FOR PARAMETRIZED REIDEMEISTER TORSION 13 One could avoid the finiteness assumption in the following result by working throughout in a convenient category of topological spaces in the sense of Steen* *rod [15]. Lemma 5.1. Suppose C has the property that each overcategory C=C is finite. Then there is a natural map |N(Nat (C=?, G))| ! holimC|N(G)|. Proof.To begin, recall that holimC|N(G)| is itself an equalizer of two maps Y Y Map (|N(C=C)|, |N(G(C))|) ! Map(|N(C=C)|, |N(G(D))|). C2C ':C!D2C These maps are constructed as with and , using the maps |N(G('))|* and |N(C=C)|*. Now, we will describe a collection of maps |N(F un(C=C, G(D)))| ! Map (|N(C=C)|, |N(G(D))|) which are natural in the sense that they send the map ~ described above to the map induced by the maps |N(C=C)|* and similarly for . There is an evaluation functor e : F un(C=C, G(D)) x C=C ! G(D) whose nerve defines a morphism N(e) : N(F un(C=C, G(D))) x N(C=C) ! N(G(D)) since the nerve functor commutes with products. Now, apply geometric realization which commutes with finite products to yield a natural map |N(e)| : |N(F un(C=C, G(D)))| x |N(C=C)| ! |N(G(D))| which is adjoint to the required map. Notice the adjoint map exists because the finiteness assumption implies |N(C=C)| is a compact Hausdorff space. Since geometric realization does not commute with arbitrary products, we must be careful that the maps |N(e)| described above induce a map from the equalizer of the pair ~, ~ rather than |N( )| or |N( )|. This gives eq(~ , ~) ! holimC|N(G)|. Fortunately, there is still a natural map eq(|N( )|, |N( )|) ! eq(~ , ~) given by the universal property of the products involved in defining the target* *. The composition is the required natural map. By Lemma 5.1, the composite displayed above yields a map |N(D)| ! holimC|N(G)|. This is the map usually called a Thomason homotopy limit problem map. The case of interest for us is when C = Simp (B)and G : Simp (B)! Cat is defined by oe 7! w Retfdp-1(oe). In particular, notice our assumption on B from the introduction implies C satisfies the assumption in the statement of Lemma 5* *.1. Notice the natural group completion map |N(G(oe))| ! A p-1(oe) yields a natural 14 W. DORABIALA AND M. W. JOHNSON ` E ' map holimC|N(G)| ! # p as well. Precomposing with our Thomason homo- B ` ' E topy limit problem map, this gives a map |N(D)| ! # p. For the category B D = w Retfd(p)and the functor w Retfd(p)! Nat (C=?,`G)discussed'in detail be- E low, this will yield a map |N(w Retfd(p))| ! # p whose target is an infinite B loop space by construction.`Thus,'we can extend over the natural group completi* *on E map |N(D)| ! A # p to build the desired coassembly map B `E ' ` E ' ffi : A # p ! # p. B B In order to complete the definition of the coassembly map ffi it suffices to de* *scribe the relevant functor F : w Retfd(p)! Nat (C=?, G). We will describe this relationsh* *ip by saying ffi is the group completion of the Thomason homotopy limit problem map associated to F. We require two other general results in this context. Lemma 5.2. (1) Suppose ' is a natural transformation from G to G0. Then the induced maps make the following diagram commute. |N(Nat (C=?, G))|___//|N(Nat (C=?, G0))| | | | | fflffl| fflffl| holimC|N(G)|________//_holimC|N(G0)| (2) There is a pseudo-product in Nat (?, !)which makes the following diagram commute. |N(Nat (C1=?, G1))| x |N(Nat (C2=?,_G2))|//_|N(Nat (C1=? x C2=?, G1 x G2))| | | | | fflffl| fflffl| holimC|N(G1)| x holim|N(G2)|___________//holim|N(G1)| x |N(G2)| 1 C2 C1xC2 Lemma 5.3. Suppose F : D ! Nat (C=?, G)and F0 : D ! Nat (C=?, G)are functors together with a natural transformation _ : F ! F0. Then the two Thomason homotopy limit problem maps |N(D)| ! holimC|N(G)| are naturally homotopic, as are their group completions. Proof.The point here is that |N(_)| induces the necessary homotopy between the two maps |N(F)| and |N(F0)| : |N(D)| ! |N(Nat (C=?, G))|. The remaining exten- sions by the composite |N(Nat (C=?, G))| ! eq(~ , ~) ! holimC|N(G)| will then be homotopic as well, so the group completions will also be homotopic. PRODUCT THEOREM FOR PARAMETRIZED REIDEMEISTER TORSION 15 Since Nat (C=?, G)is defined as an equalizer of and inQCat, defining a fu* *nctor into Nat (C=?, G)is equivalent to defining a functor into C2CF un(C=C, G(C)) * *so that the two composite functors Y D ! F un(C=C, G(D))' ':C!D2C agree. As each of these target categories is a product, the universal propertie* *s imply it suffices to check on each projection. In other words, we should define a ser* *ies of functors FC : D ! F un(C=C, G(C)) such that each diagram FC D _______________________//_F un(C=C, G(C)) FD || |G(')*| fflffl| fflffl| F un(C=D, G(D))______(C=')*____//_F un(C=C, G(D))' commutes. In our specific situation, this means we need to define functors -1 Foe: w Retfd(p)! F un(Simp (B)=oe, w Retfdp (oe)). The functors we have in mind send X ø E to the functor ffi ffi* -1 ffi 7! ioe*(iB ) (X) ø p (oe). Functoriality of this assignment requires a natural weak equivalence ffi ffi* ø ø * ioe*(iB ) (X) ! ioe*(iB ) (X) whenever ffi ø. However, uniqueness of inclusions implies ffi* ø * i ffij* iø iB (X) = iB (X) and ø ffi ffi ioe* iø *= ioe* so ffi ffi* ø ffi ffi*ø * ioe*(iB ) (X) = ioe*iø * iø iB (X). Hence, iøoe*applied to the unit of adjunction gives the required natural weak equivalence. Clearly, this is natural with respect to further inclusions in ei* *ther direction. Notice that C=C in this case simply consists of the subcomplexes of oe, and t* *he functors C=' for ' : ø ! oe simply become the inclusion of the subcomplexes of ø as subcomplexes of some larger simplex oe. Hence the associated functor -1 fd -1 G(') : w Retfdp (ø) ! w Ret p (oe) should be the pushforward functor iøoe*. 16 W. DORABIALA AND M. W. JOHNSON Now the required commutative diagrams Fø fd -1 w Retfd(p)________________//_F un(Simp (B)=ø, w Ret p (ø) ) | Foe|| | fflffl| || F un(Simp (B)=oe, w Retfdp-1(oe)) |((iøoe)*)* XXXXX ø* | X(Simp(B)=ioe)XXXXXX | XXXXXXX | XX++X |fflffl F un(Simp (B)=ø, w Retfdp-1(oe))iøoe are reduced to the statement that ø i ffii ffij* j ffii ffij* ioe* iø* iB (X) = ioe* iB (X). Thus, our assignment gives the required functor F : w Retfd(p)! Nat (C=?, G). If we choose a different initial functor G, sending oe`to w Retfd,cyp-1(oe), * *the E ' ` E ' same process yields the cyclic coassembly map fficy: Acy # p ! cy # p as the B B group completion of the Thomason homotopy limit problem map of a modification of the functor F above. Similarly, one can alter G to send oe to w RetfdRp-1(oe) and notice the functor F described above also yields a functor F : w RetfdR(p)! Nat (C=?, G) in this case. Thus, we have another coassembly map ` E ' ` E ' ffiR : AR # p ! R # p . B B In order to define the Reidemeister torsion, we needed the existence of a can* *onical path fi from V(Ø (p)) to the basepoint in holimK (R). Using the machinery of t* *his section, it is straightforward to describe this path. To begin, notice any object X of Retfd,cy(p)sitting in RetfdR(p)has the prope* *rty that the retraction is a weak equivalence by the definition of Retfd,cy(p). Hen* *ce, there is a natural map from X to E in`w RetfdR(p), which leads to a natural path E ' from the point associated to X in AR # p to the basepoint. B i j Now suppose X represents an object in Nat C=?, w RetfdRp-1(?)which comes i j from an object in Nat C=?, w Retfd,cyp-1(?)under the "inclusion" functor. i j There is another object of Nat C=?, w RetfdRp-1(?)which consists of the con- stant functors to p-1(?) and serves as the äb sepoint". Once again, the combina* *tion of all of the relevant retractions will provide a natural morphism`from'X to p-* *1(?). E Since we will choose the image of p-1(?) as a point of R # p as the basepoin* *t, ` ' B E this gives a natural path fi0 in R # p from the point associated to X to the B basepoint. PRODUCT THEOREM FOR PARAMETRIZED REIDEMEISTER TORSION 17 Our choice for X will`be the'image`of'E t E under the functor which builds E E the coassembly map Acy # p ! cy # p , which is only possible if the bundle B B ` ' E involved is an acyclic flat bundle . Then fi0gives a natural path in R # p f* *rom B the image of Øcy(p) to the basepoint, since Øcy(p) was defined as the image of ØAcy(p)(which itself corresponds to the retractive space E t E) under the cyclic coassembly map. We define our natural path fi in holimK (R) from V(Ø (p)) to the basepoint as V applied to the the path fi0. Alternatively, we could define fi by a construction quite similar to that of * *fi0, using Nat (C=?, wCh*(R) )which corresponds to the holim of the constant functor K (R). Since the postcomposition by an exact functor preserves the zero map, the natural map from X to the äb sepoint" will be sent to the (unique) natural map from the image of X to the constant functor on the zero chain complex, which se* *rves as the basepoint in Ch*(R) . Taking group completions then says this definition of fi agrees with that given above. The interested reader may also see the path given by [5], Observation 6.4, along with Propositions 6.6 and 6.7 is yet anoth* *er description of fi. 6. Products and Natural Maps The purpose of this section is to define the multiplications referred to prev* *iously in full detail. First, recall the definition of ~A as induced by the external smash product of retractive spaces defined in Section 3. In fact, this defines a bi-exact functo* *r, hence a natural map `E1 ' ` E2 ' ` E1xE2 ' ~A : A # p1x A # p2! A # p1x p2 B1 B2 B1xB2 which clearly descends to the smash product by construction to give `E1 ' ` E2 ' ` E1xE2 ' ~A : A # p1^ A # p2! A # p1x p2. B1 B2 B1xB2 We would like to use the external smash product again to define `E1 ' ` E2 ' ` E1xE2 ' ~Acy: A # p1^ Acy # p2 ! Acy # p1x p2 . B1 B2 B1xB2 However, it is not yet clear that X^E1xE2Y lies in the subcategory Retfd,cy(p1 * *x p2). In other words, the construction yields a map ` E1 ' ` E2 ' ` E1xE2 ' A # p1 ^ Acy # p2 ! A # p1x p2 B1 B2 B1xB2 which currently has the wrong target to be our ~Acy. One should also keep in mi* *nd that we need flat bundles OE1 : V1 ! E1 and OE2 : V2 ! E2 in order to even defi* *ne the spaces involved in ~Acy. To deal with this problem we first need a technic* *al result. When describing relative chain complexes with local coefficients, the symbol C*(A, B; V )will indicate that V is a flat bundle over B, A is retractive over* * B and the bundle in question over A is the pullback of V over the retraction, whi* *ch remains a flat bundle. Also, given OE1 : V1 ! E1 and OE2 : V2 ! E2, the symbol 18 W. DORABIALA AND M. W. JOHNSON V1^ V2 will denote the tensor product of the two bundles over E1 x E2 given by pulling back each Vi over the relevant projection map. Lemma 6.1. Suppose X 2 RetfdR1(p1). Then the functor X^E1xE2? : RetfdR2(p2)! RetfdR1 AR2(p1 x p2) preserves weak equivalences. Proof.Suppose f : Y ! Z is a weak equivalence in RetfdR2(p2). This means f induces a quasi-isomorphism C*(Y, E2, V2)! C*(Z, E2, V2) with local coefficients. Thus, tensoring with C*(X, E1, V1)yields another quas* *i- isomorphism C*(X, E1, V1) C*(Y, E2, V2)! C*(X, E1, V1) C*(Z, E2, V2) since V1 is a flat bundle. However, the relative Eilenberg-Zilber Theorem with * *local coefficients (exercise 8 on page 282 of [14] or [6]) then implies the existence* * of a quasi-isomorphism * * C* X x Y, X x E2 [ E1 x Y, r1(V1^ V2)! C* X x Z, X x E2 [ E1 x Z, r2(V1^ V2) with r1 : X x E2 [ E1 x Y ! E1 x E2 and similarly for r2. Using the relative Mayer-Vietoris sequence with local coefficients and compact supports (see page 412, exercise 8 of [3]) together with the fact that X x E2 [ E1 x Y ! X x Y is a cofibration, one concludes that there is a natural quasi-isomorphism * C* X x Y, X x E2 [ E1 x Y, r1(V1^ V2) C* X ^E1xE2 Y, E1 x E2, V1^ V2 and similarly for Z. Thus, transitivity implies the map f induces a quasi-isomo* *rphism C* X ^E1xE2 Y, E1 x E2, V1^ V2! C* X ^E1xE2 Z, E1 x E2, V1^ V2 as well. Lemma 6.2. Suppose X 2 Retfd(p1)and Y 2 Retfd,cy(p2). Then X ^E1xE2 Y 2 Retfd,cy(p1 x p2). Proof.The statement that Y 2 Retfd,cy(p2)is equivalent to saying Y 2 RetfdR2(p2) with Y ! E2 a weak equivalence in this structure. Then Lemma 6.1 implies X ^E1xE2 Y ! X ^E1xE2 E2 E1 x E2 is a weak equivalence in RetfdR1,2(p1 x,p2)or equivalently that X ^E1xE2 Y 2 Retfd,cy(p1 x p2). Lemma 6.3. (1) There is a natural external multiplication `E1 ' ` E2 ' ` E1xE2 ' ~AR : AR1 # p1^ AR2 # p2! AR1 AR2 # p1x p2 B1 B2 B1xB2 PRODUCT THEOREM FOR PARAMETRIZED REIDEMEISTER TORSION 19 (2) The following diagram commutes. ` E1 ' ` E2 ' ` E1xE2 ' A # p1^ A # p2 __~A___//_A # p1x p2 B1 B2 B1xB2 | | | | ` ' fflffl|` ' ` fflffl|' E1 E2 E1xE2 AR1 # p1^ AR2 # p2 _~__//_AR # p1x p2 B1 B2 AR B1xB2 Proof.The multiplication ~AR is induced by the same external smash product ^E1xE2 : RetfdR1(p1)x RetfdR2(p2)! RetfdR(p1 x p2). With the homology chain equivalences as weak equivalences, we must show this remains a bi-exact functor. Since we have not changed any other portion of the Waldhausen structure, we only need to check that fixing an element X 2 RetfdR1(* *p1), X^E1xE2? preserves the new class of weak equivalences (and similarly for ?^E1xE2 Y ). However, this follows immediately from Lemma 6.1. Once we know the same functor induces the multiplication, the commutativity of the diagram follows from the fact that the vertical maps are induced by the ide* *ntity functor at the level of retractive spaces. The product ` E1 ' ` E2 ' ` E1xE2 ' _ : # p1^ # p2 ! # p1x p2 B1 B2 B1xB2 is defined as the following composition: -1 -1 holim A p1 (oe1)^ holim A p2 (oe2) oe12Simp(B1) oe22Simp(B2) | | fflffl| holim A p-11(oe1)^ A p-12(oe2) oe1xoe22Simp(B1xB2) (~A)*|| fflffl| holim A (p1 x p2)-1(oe1 x oe2). oe1xoe22Simp(B1xB2) where the first map comes from the interaction of holim and products. Suppose p1 is a perfect fibration with flat bundle V1 and p2 is a perfect fib* *ration with acyclic flat bundle V2. We can now describe ` E1 ' ` E2 ' ` E1xE2 ' _cy: # p1^ cy # p2 ! cy # p1x p2 B1 B2 B1xB2 20 W. DORABIALA AND M. W. JOHNSON similarly as the following composition. -1 -1 holim A p1 (oe1)^ holim Acy p2 (oe2) oe12Simp(B1) oe22Simp(B2) | | fflffl| holim A p-11(oe1)^ Acy p-12(oe2) oe1xoe22Simp(B1xB2) (~Acy)*|| fflffl| holim Acy (p1 x p2)-1(oe1 x oe2) oe1xoe22Simp(B1xB2) ` E ' `E ' Next is the definition of ffl : cy # p ! Wh RB # p. We begin with a lemma B B from the non-parametrized case. Lemma 6.4. The linearization map factors through A R(E) and its generalized linearization map. That is, the following diagram commutes. A (E) ____//_AR(E) II III ~V| ~V III$$Ifflffl|| K (R) See [4] for details. Now, consider the following diagram which commutes as a result of Lemma 6.4. = holimA p-1(oe)_____//_holimAp-1(oe) | | | | V fflffl| fflffl| holimAR (p-1(oe))_V___//_holimK(R) As mentioned previously, Acy p-1(oe) is homotopy equivalent to the homotopy fiber of -1 -1 A p (oe)! AR p (oe). ` E* * ' Since homotopy fibers commute with homotopy inverse limits, this implies cy #* * p B is homotopy equivalent to the homotopy fiber`of the'left vertical map in the pr* *e- E vious diagram. However, by definition Wh RB # p is the homotopy fiber of the B right vertical in the previous`diagram.'Thus,`commutativity'of the diagram above E E induces a map ffl : cy # p ! Wh RB # p. B B We now need a technical lemma. PRODUCT THEOREM FOR PARAMETRIZED REIDEMEISTER TORSION 21 Lemma 6.5. (1) Given a diagram in T op f T ^ W _____//Y 1^p|| |q| fflffl| fflffl| T ^ X _g___//Z together with a pointed homotopy H from qf to g(1 ^ p) there is a natural map 'H making the following extension commute: T ^ hofiberp'H__//hofiberq | | | | fflffl| fflffl| T ^ W __________//Y. (2) Suppose one has a diagram in T op T ^ M _e___//N 1^i|| |j| fflffl|f fflffl| T ^ W _____//Y 1^p|| |q| fflffl| fflffl| T ^ X _g___//Z where je = f(1 ^ i) and H is a pointed homotopy as above. Then one has a homotopy commutative diagram T ^ hofiberi______//_hofiberj | | | | fflffl| fflffl| T ^ hofiber(pi)'H//_hofiber(qj). Proof.Recall, a point in hofiberp consists of a pair (ff, w) where w 2 W and ff : I ! X is a path from p(w) to the basepoint. Define 'H (t, ff, w) as the p* *air (g(t, ff)H(t, p(w), ?), f(t, w)), where the first component means the concatena* *tion of these two paths in Z. For the second statement, define the map T ^ hofiberi ! hofiberj by a simpler variation of 'H , namely (t, ff, m) is mapped to (f(t, ff), e(t, m* *)). (No- tice this is homotopic to the variation of 'H using a constant homotopy.) In or* *der to establish the homotopy commutativity of the diagram, it suffices to show tha* *t the paths qf(t, ff) and g(t, p(ff))H(t, i(m), ?) are homotopic relative to the endp* *oints. If we use s to denote the time variables for ff and H, and r as a time variable* * for our homotopy, the relevant formula is: ( 1 _; (3) K(s, r) = H(t, ff(s - rs), 2rs),if s 2 1 H(t, ff(s - r + rs), r),if s _2. 22 W. DORABIALA AND M. W. JOHNSON The multiplication ` E1 ' ` E2 ' ` E1xE2 ' : # p1^ Wh R2B2# p2 ! Wh R1BAR21xB2 # p1x p2 B1 B2 B1xB2 is defined by applying Lemma 6.5 to the homotopy commutative diagram given by the following proposition. Proposition 6.6. (1)There is a natural commutative diagram as below: ` E1 ' ` E2 ' ` E1xE2 ' # p1 ^ # p2 _________//_ # p1x p2 B1 B2 B1xB2 | | | | ` ' fflffl|` ' ` fflffl| ' E1 E2 E1xE2 # p1 ^ R2 # p2_____//R1 AR2 # p1x p2. B1 B2 B1xB2 (2) There is a natural choice of homotopy between the two compositions in the following diagram: ` E1 ' ` E2 ' ` E1xE2 ' # p1 ^ R2 # p2_____//R1 AR2 # p1x p2 B1 B2 B1xB2 1^ V || |V| ` ' fflffl| | E1 fflffl| # p1 ^ holimK (R2)______//_holimK(R1 A R2). B1 Proof.We begin with the diagram -1 -1 A p1 (oe1)^ A p2 (oe2)_________//A(p1 x p2)-1(oe1 x oe2) | | | | fflffl| fflffl| AR1(p-11(oe1)) ^ AR2(p-12(oe2))//_AR1 AR2((p1 x p2)-1(oe1 x oe2)) which commutes by Lemma 6.3. Taking holimover the category Simp (B1 x B2) then implies the following diagram commutes -1 -1 holim A p1 (oe1)^ A p2 (oe2) WW | WWWWWWW | WWWWWWW || WWWWW++ | holimA (p1 x p2)-1(oe1 x oe2) | | | | | fflffl| || holim AR1(p-11(oe1)) ^ AR2(p-12(oe2)) | WWWWW | WWWWWWW | WWWWWW | WWW++ fflffl| holimAR1 AR2 ((p1 x p2)-1(oe1 x oe2)). PRODUCT THEOREM FOR PARAMETRIZED REIDEMEISTER TORSION 23 Of course, one also has a commutative diagram associated to holim of products -1 holim A p1 (oe1) Simp(B1) -1 -1 ^ ______//_holimSimp(B1xB2)Ap1^(oe1)Ap2 (oe2) holim A p-12(oe2) Simp(B2) | | || | | fflffl| -1 | Sholimimp(B1)AR1(p1 (oe1)) fflffl| -1 -1 ^ _____//Sholimimp(B1xB2)AR1(p1 (oe1)) ^ AR2(p2 (oe2)) holim AR2(p-12(oe2)) Simp(B2) which taken together yields a commutative diagram which we reinterpret as the following ` E1 ' ` E2 ' ` E1xE2 ' # p1^ # p2 ____________________________// # p1x p2 B1 B2 B1xB2 | | | | ` ' fflffl|`' ` fflffl| ' E1 E2 E1xE2 # p1^ R2 # p2 R1 AR2 # p1x p2. B1 B2P 77 B1xB2 PPPP nnnnn PPPP nnnn PP''P`' ` nnn' E1 E2 R 1 # p1 ^ R2 # p2 B1 B2 For the second diagram, begin with the following diagram (coming from the non-parametrized case) which commutes up to a natural homotopy -1 -1 ~A A p1 (oe1)^ AR2(p2 (oe2))___//_AR1 AR2((p1 x p2)-1(oe1 x oe2)) ~V1^~R2V2|| |~R1VAR21|V2 fflffl| fflffl| K(R1) ^ K(R2) _______jK________//_K(R1 A R2) where the bottom map is Loday's pairing on K-theory [13]. The natural homotopy in this case comes from the Eilenberg-Zilber map on chain complexes. Now take holimover the category Simp (B1 x B2)and proceed as above, keeping in mind that the upper left corner in each diagram is not symmetric. The statement about naturality of the homotopy then follows from the naturality of the Eilenberg-Zi* *lber map. We can now prove the second statement of Proposition 2.2. 24 W. DORABIALA AND M. W. JOHNSON Proposition 6.7. The diagram ` E1 ' ` E2 ' _cy ` E1xE2 ' # p1 x cy # p2 _________//_ cy # p1x p2 B1 B2 B1xB2 1xffl2|| ffl1,2|| ` ' fflffl|` ' ` fflffl| ' E1 E2 R R E1xE2 # p1 x Wh R2B2# p2 _____//Wh1BA12xB2 # p1x p2. B1 B2 B1xB2 commutes up to a natural homotopy. Proof.The homotopy commutative square comes from applying Lemma 6.5 to the diagrams provided by Proposition 6.6. We can now give a proof of Proposition 2.1. Proof of Proposition 2.1.Consider the diagram -1 = -1 holimA p2 (oe2)______//holimAp2 (oe2) | | | |V fflffl| fflffl| holimAR2(p-12(oe2))_V___//holimK(R2) ` * *E2 ' involved in defining ffl. By definition, øR2 (p2)is a lift of Ø (p2)to Wh R2B2* * # p2 * *B2 associated to a specific choice of path fi from V(Ø (p2)) to the basepoint, di* *scussed at the end of section 5. It is important to keep in mind this path arises as th* *e image of a similar path to the basepoint fi0 in each A R2(p-12(oe2)), given by effect* *ively the same construction.` Since'the`pair'(Øcy(p), fi0) is the image of Øcy(p)in t* *he E E homotopy fiber F of # p ! R # p , the composite B B ` E ' ` E ' # p ! F ! R # p B B will send Øcy(p)to the point in the Whitehead space corresponding to the pair (Ø (p), fi). However, this point was our definition of the Reidemeister torsion* * class øR2 (p2). 7.The coassembly map is Multiplicative The purpose of this section is to prove that the diagram ` E1 ' ` E2 ' ` ' cy cy E1xE2 (4) A # p1^ Acy # p2 _~A__//A # p1x p2 B1 B2 B1xB2 ffi1xfficy2|| fficy1,2|| ` ' fflffl|`' ` fflffl|' E1 E2 E1xE2 # p1^ cy # p2 __cy_// cy # p1x p2. B1 B2 _ B1xB2 is commutative up to a natural homotopy. We begin with the proof of Lemma 2.5. PRODUCT THEOREM FOR PARAMETRIZED REIDEMEISTER TORSION 25 Proof of Lemma 2.5.Part 1 follows from the fact that the multiplications ~A and ~Acyare both defined by the same bi-exact functor at the level of retractive sp* *aces. To see part 2, recall that Retfd,cy(p)! Retfd(p)is the inclusion of a sub- Waldhausen category. Hence, wS. Retfd,cy(p)! wS. Retfd(p) is the inclusion of a sub-simplicial category. This implies ` E ' `E ' Acy # p = |wS.Retfd,cy(p)| ! |wS.Retfd(p)| = A # p B B is an injection. Finally, part 3 follows from the nonparametrized analogue`of'part`2 and'the E E standard model for the homotopy inverse limit. Since Acy # p ! A # p is an B B injection, it should be clear that for each oe 2 Simp(B) one has -1 -1 Map (| Simp(B) =oe|, Acy p (oe)) ! Map (| Simp(B) =oe|, A p (oe)) an injection as a postcomposition by an injection. This implies the product of such maps is also an injection. However, the standard model for the homotopy inverse limit is the appropriate subspace of such a product and the restriction* * of an injection remains an injection. As a consequence of Lemma 2.5, it should be clear that the first portion of Proposition 2.2 is a corollary of Theorem 2.3, where we dealt with the product rather than smash product versions of ~A and _ for technical reasons. We begin working toward this proof with another technical lemma. Lemma 7.1. The product map `E1 ' ` E2 ' |w Retfd(p1)| x |w Retfd(p2)| ! A # p1x A # p2 B1 B2 retains the group completion property. Proof.The definition of the S. construction implies that for Waldhausen categor* *ies C and D one has a natural isomorphism S.(C x D) S.(C) x S.(D). Of course, this assumes the Waldhausen structure on C x D is that coming from the product. Clearly, the inclusion i j i j |w Retfd(p1)x Retfd(p2)| ! |wS. Retfd(p1)x Retfd(p2)| has the group completion property by [16]. However, the previous paragraph leads us to conclude the target is isomorphic to `E1 ' ` E2 ' |wS.(Retfd(p1)) x wS.(Retfd(p2))| A # p1x A # p2. B1 B2 On the other hand, we also have an isomorphism i j |w Retfd(p1)| x |w Retfd(p2)| |w Retfd(p1)x Retfd(p2)|. 26 W. DORABIALA AND M. W. JOHNSON Thus, it suffices to notice that the composite of these maps `E1 ' ` E2 ' |w Retfd(p1)| x |w Retfd(p2)| ! A # p1x A # p2 B1 B2 is the product of the maps i j `Ei ' |w Retfd(pi)| ! |wS. Retfd(pi)| = A # pi. Bi There are now two functors i j w Retfd(p1)x w Retfd(p2)! Nat Simp(B1 x B2)=?, w Retfd((p1 x p2)-1(?). The functor F1 comes from following the external smash product (which induces the external product in parametrized A-theory) ^E1xE2 : w Retfd(p1)x w Retfd(p2)! w Retfd(p1 x p2) with the functor i j w Retfd(p1 x p2)! Nat Simp(B1 x B2)=?, w Retfd(p1 x p2)-1(?) described after Lemma 5.3. The second functor F2 comes from first taking the product of the functors i j w Retfd(pi)! Nat Simp(Bi)=?, w Retfdp-1i(?) as above, followed by a functor i j Nat Simp(B1)=?, w Retfdp-11(?) i x j Nat Simp(B2)=?, w Retfdp-12(?) | | i fflffl| j Nat Simp(B1 x B2)=?, w Retfd(p1 x p2)-1(?) given by the pseudo-product in Nat (?, !)followed by the functor induced by con- sidering the external smash product as a natural transformation. Lemma 7.2. (1) The composite ffi1,2(~A) from Theorem 2.3 is the group com- pletion of the Thomason homotopy limit problem map associated to F1. (2) The composite _ (ffi1x ffi2) from Theorem 2.3 is the group completion of* * the Thomason homotopy limit problem map associated to F2. Proof.For the first claim, recall that ffi1,2is the group completion of the sec* *ond functor in the definition of F1 and ~A is the group completion of the external smash product. PRODUCT THEOREM FOR PARAMETRIZED REIDEMEISTER TORSION 27 To verify the second claim, it suffices by the uniqueness of group completion* *s to establish commutativity of the following diagram ` E1 ' ` E2 ' |w Retfd(p1)| x |w Retfd(p2)|__________//A # p1 x A # p2 | B1 B2 | | | ffi1xffi2| i fflffl| j | |Nat Simp(B1)=?, w Retfdp-11? | ` ' fflffl|`' E1 E2 i x j___________//_ # p1 x # p2 |Nat Simp(B2)=?, w Retfdp-12? | B1 B2 | | | _| | | i fflffl| j ` E1xEfflffl|2' |Nat Simp(B1 x B2)=?, w Retfd(p1 x p2)-1?|_______// # p1x p2. B1xB2 The commutativity of the top square follows from the fact that group completions commute with products by naturality. Thus, it remains only to show the bottom square commutes. However, by the construction of F2, this bottom square itself factors as the group completion of a pair of squares, i j |Nat Simp(B1)=?, w Retfdp-11? | ` E1 ' ` E2 ' i x j___________// # p1 x # p2 |Nat Simp(B2)=?, w Retfdp-12? | B1 | B2 | || | | i fflffl| j | fd -1 fd -1 | |Nat Simp(B1 x B2)=?, w Ret p1 ? x w Ret p2 ? | | UUUU | UUUUU | UUUUU | UUUUUU** |fflffl holim A p-11(oe1) Simp(B1xB2) x A p-12(oe2) and i j |Nat Simp(B1 x B2)=?, w Retfdp-11?x w Retfdp-12? | | U | UUUUU | UUUUU | UUUUU | UUUU**holim A p-1(oe1) | Simp(B1xB2) 1 | x | -1 | A p2 (oe2) | | | | | i fflffl| j ` E1x|fflfflE2' |Nat Simp(B1 x B2)=?, w Retfd(p1 x p2)-1?|_______// # p1x p2. B1xB2 28 W. DORABIALA AND M. W. JOHNSON each of which commutes by Lemma 5.2. The first comes from the pseudo-product in Nat (?, !)and the second from the map induced by the external smash product considered as a natural transformation. Proposition 7.3. There is a functor i j F0 : w Retfd(p1)xw Retfd(p2)! Nat Simp(B1 x B2)=?, w Retfd(p1 x p2)-1(?) together with natural transformations (_n) : F0 ! Fn for n = 1,2. Proof.We return to the notation of section 5 in order to describe F1 and F2 alo* *ng with their effect on an object (X1iø E1,jX2*ø E2) of w Retfd(p1)xiwjRetfd(p2). To simplify notation, let (in)* = ioenBnand let (jn)* = iffinoen. First, in b* *uilding * F1 one sends this pair to the functor (ffi1, ffi2) 7! (j1 x j2)*(i1 x i2)*(X1 ^E1xE2 X2). Next, in building F2 one sends the pair to the functor (ffi1, ffi2) 7! (j1)*(i1)*(X1) ^p-11(oe1)xp-12(oe2)(j2)*(i2)*(X2). The new functor F0 is defined by i j (ffi1, ffi2) 7! (j1 x j2)* (i1)*(X1) ^p-11(ffi1)xp-12(ffi2)(i2)*(X2). The natural transformation _1 is built from the natural weak equivalences which arise by applying (j1xj2)* to the natural weak equivalence of type (1) in Lemma* * 3.4. Similarly, the natural transformation _2 is built from the natural weak equival* *ences of type (2) in Lemma 3.4. This suffices to allow us to prove Theorem 2.3. Proof of Theorem 2.3.Lemma 5.3 together with Proposition 7.3 imply the Thoma- son homotopy limit problem maps associated to F1 and F2 are naturally homo- topic (via the Thomason homotopy limit problem map associated to F0). However, Lemma 7.2 then implies ` E1 ' ` E2 ' ` E1xE2 ' A # p1 x A # p2__~A_//A # p1x p2 B1 B2 B1xB2 ffi1xffi2|| ffi1,2|| ` ' |fflffl`' ` fflffl|' E1 E2 E1xE2 # p1 x # p2 _____// # p1x p2. B1 B2 _ B1xB2 commutes up to a natural homotopy. 8.Open Questions There are two main directions (aside from the general goal mentioned in the introduction) in which we would like to proceed in the future. The first would * *be to give a definition of smooth parametrized torsion after [5]. This would lead * *to an appropriate definition of higher Reidemeister torsion as in [8]. Specifically, * *Theorem 6.6.1 of [8] and possibly conjecture 6.6.7 would follow from the generalization* * of Theorem 1.4 to higher torsion. PRODUCT THEOREM FOR PARAMETRIZED REIDEMEISTER TORSION 29 To see Theorem 6.6.1 of [8], consider the product of a perfect fibration p1 w* *ith the constant map p2 : N ! * for a manifold N. Then Theorem 1.4 gives (Ø (p2), øR1 (p1)) ' øR1 AR2 (p1 x p2). However, from [5] we know Ø (p2)= Ø (N), the standard Euler characteristic of t* *he manifold N. The second logical direction is to pursue a definition of Reidemeister torsion which does not require the acyclicity assumption. A careful analysis of the res* *ults in this paper suggest the effective role of the acyclicity assumption is to est* *ablish E t E as a multiplicatively natural element of Retfd,cy(p). Without the acycli* *c- ity assumption, one might still hope to construct such a multiplicatively natur* *al element X(p) in Retfd,cy(p). In that case, the element X(p) could play the role of X in the discussion at the end of Section 5, giving an analog of the path fi* *(p) associated to X(p). 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In Algebraic and geomet* *ric topol- ogy (New Brunswick, N.J., 1983), volume 1126 of Lecture Notes in Math., page* *s 318-419. Springer, Berlin, 1985. [17]Bruce Williams. Bivariant Riemann Roch theorems. In Geometry and topology: * * Aarhus (1998), volume 258 of Contemp. Math., pages 377-393. Amer. Math. Soc., Provi* *dence, RI, 2000. Institute of Mathematics, Szczecin University, ul. Wielkopolska 15, 70-451 Sz* *czecin, Poland Current address: Department of Mathematics, Penn State Altoona, Altoona, PA 1* *6601-3760 E-mail address: wud2@psu.edu Department of Mathematics, Penn State Altoona, Altoona, PA 16601-3760 E-mail address: mwj3@psu.edu